In general, the complexity of a formulation to approximate the radiation boundary condition is a function of the degree of space-time non-locality introduced in the boundary model. In the case of classical FDs and FEs, non-locality is an unnatural condition: Consequently, in order to obtain effective solutions, the analyst must sacrifice accuracy at the expense of non-locality; with accurate results obtained after a trade-off between non-locality and the use of Saint-Venant-end-effects leading to large computational models. One of the first contributions in absorbing boundaries, but still one of the most popular ones, is the fully-local, viscous boundary element by [1]}. This boundary provides the conceptual basis of most of the currently used elements, which exhibit various levels of efficiency with increasing levels of non-locality, [2]}, [3]}, [4]}, [5]}, [6]}. Moreover, many of these absorbing conditions have been used in recent FDs and FEs simulations of seismic regions, as large as the state of California and for frequencies up to \(4.0\)  Hz, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}. In a second class of methods, which are inherently non-local in space and time, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, the radiation condition is imposed through fully coupled boundary integral formulations: In those cases the half-space condition is analytically satisfied by the Green's function, but the resulting algorithm, although highly accurate is also computationally demanding with coefficient matrices which are non-symmetric and fully populated.
